Dic: deep (in some applied meanings)

6. ADJ : usu ADJ n △You use deep to emphasize the seriousness, strength, importance, or degree of something. (= profound)

• I had a deep admiration for Sartre.
• He wants to express his deep sympathy to the family. 

CF. serious △10 extreme or serious.

• He's in deep trouble.
• a deep economic recession
• The affair had exposed deep divisions within the party.
• a place of great power and of deep significance

* * *

14. ADJ △If you describe something such as a problem or a piece of writing as deep, you mean that it is important, serious, or complicated.

• They're written as adventure stories. They're not intended to be deep.

CF. knowledge △12. showing great knowledge or understanding.

• a deep understanding

CF. difficult to understand △13 difficult to understand (= profound)

• This discussion's getting too deep for me.
• He pondered, as if over some deep philosophical point.

* * *

9. ADJ : v-link ADJ in n △If you are deep in thought or deep in conversation, you are concentrating very hard on what you are thinking or saying and are not aware of the things that are happening around you.

• Abby had been so deep in thought that she had walked past her aunt's car without even seeing it.

CF. involved 14 deep in something △fully involved in an activity or a state

• to be deep in thought/conversation
• He is often so deep in his books that he forgets to eat.
• The firm ended up deep in debt.

......... Cobuild, OALD